Localization and delocalization of light in photonic moire lattices
Peng Wang, Yuanlin Zheng, Xianfeng Chen, Changming Huang, Yaroslav V. Kartashov, Lluis Torner, Vladimir V. Konotop, Fangwei Ye
LLocalization and delocalization of light in photonic moir´e lattices
Peng Wang, Yuanlin Zheng, Xianfeng Chen, Changming Huang, YaroslavV. Kartashov, , Lluis Torner, , Vladimir V. Konotop, Fangwei Ye ∗ School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Department of Electronic Information and Physics,Changzhi University, Shanxi 046011, China ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,08860 Castelldefels (Barcelona), Spain Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia Universitat Politecnica de Catalunya, 08034 Barcelona, Spain Departamento de F´ısica, and Centro de F´ısica Te´orica e Computacional,Faculdade de Ciˆencias, Universidade de Lisboa,Campo Grande, Ed. C8, Lisboa 1749-016, Portugal ∗ Corresponding author: [email protected] (Dated: September 18, 2020) a r X i v : . [ phy s i c s . op ti c s ] S e p oir´e lattices consist of two identical periodic structures overlaid with a rel-ative rotation angle. Observed even in everyday life, moir´e lattices have beenalso produced with coupled graphene-hexagonal boron nitride monolayers [1–3], graphene-graphene layers [4, 5], and layers on a silicon carbide surface [6].The recent surge of interest in moir´e lattices is connected with a possibilityto explore in such structures a rich variety of unusual physical phenomena,such as commensurable-incommensurable transitions and topological defects [2],emergence of insulating states due to band flattening [4, 7], unconventional su-perconductivity [5] controlled by the rotation angle between monolayers [8, 9],quantum Hall effect [10], realization of non-Abelian gauge potentials [11], aswell as quasicrystals appearing at special rotation angles [12]. However, a fun-damental question that remains unexplored is the evolution of waves in thepotentials defined by the moir´e lattices. Here we experimentally create two-dimensional photonic moir´e lattices, which, unlike their crystalline predeces-sors, have controllable parameters and symmetry allowing to explore transitionsbetween structures with fundamentally different periodicity: periodic, generalaperiodic and quasi-crystal ones. Equipped with such realization, we observelocalization of light in deterministic linear lattices. Such localization is basedon the flat-band physics [7], thus contrasting with previous schemes based onlight diffusion in optical quasicrystals [14], where disorder is required [15] forthe onset of Anderson localization [16]. Using commensurable and incommen-surable moir´e patterns, we also provide the first experimental demonstrationof two-dimensional localization-delocalization-transition (LDT) of wavepacket inan optical setting. Moir´e lattices may feature almost arbitrary geometry con-sistent with the crystallographic symmetry groups of the sublattices, they allowcomprehensive exploration of the physics of transitions between periodic andaperiodic phases and offer an alternative way for manipulating light by light. One of the most important properties of an engineered optical system is its capability toaffect the light beam in a prescribed manner, in particular to localize it or to control its2iffraction. The importance of the problem of wave localization extends far beyond optics.It was studied in all branches of physics dealing with wave phenomena. In particular, it iswell known that homogeneous or strictly periodic linear systems cannot result in localizationof waves, and that inhomogeneities, random or regular, or nonlinear effects are required forlocalization. Wave localization in a random medium, alias Anderson localization [17], is ahallmark discovery of the condensed-matter physics. All electronic states in one- and two-dimensional potentials with uncorrelated disorder are localized. Three-dimensional systemswith disordered potentials are known to have both localized and delocalized eigenstates [16],separated by an energy known as the mobility edge [18].If disorder is correlated, a mobility edge may exist in one-dimensional systems too, asit was observed experimentally for a Bose-Einstein condensate in a speckle potential [19].Furthermore, coexistence of localized and delocalized eigenstates has been predicted also inregular quasiperiodic one-dimensional systems, first in the discrete Aubri-Andr´e [20] modeland later in continuous optical and matter-wave systems [21–23]. Quasiperiodic (or aperi-odic) structures, even those that possess long-range order, fundamentally differ both fromperiodic systems where all eigenmodes are delocalized Bloch waves, and from disorderedmedia where all states are localized (in one or two dimensions). Upon variation of the pa-rameters of a quasiperiodic system, it is possible to observe transition between localized anddelocalized states. Such LDT has been observed in one-dimensional quasiperiodic optical [24]and in atomic systems [25, 26].Wave localization is sensitive to dimensionality of the problem, irrespective of the type ofthe medium. Anderson localization and mobility edge in two-dimensional disordered systemswere first reported in the experiment with bending waves [27] and later in optically induceddisordered lattices [28]. In quasicrystals localization has been observed only under the actionof nonlinearity [14] and in the presence of strong disorder [15]. Localization and delocalizationof light in two-dimensional systems without any type of disorder and nonlinearity, have neverbeen observed so far.Here we report on the first experimental realization of reconfigurable photonic moir´e lat-tices with controllable parameters and symmetry. The lattices are induced by two super-3mposed periodic patterns [29, 30] (sublattices) with either square or hexagonal primitivecells [31]. The sublattices of each moir´e pattern have tunable amplitudes and twist an-gle. Depending on the twist angle a photonic moir´e lattice may have different periodic(commensurable) structure or aperiodic (incommensurable) structure without translationalperiodicity, but they always feature the same rotational symmetry as the symmetry of thesublattices. Moir´e lattice can also transform into quasicrystals with higher rotational sym-metry (e.g. the case of the Stampfli pattern [12]). The angles at which a commensurablephase (periodicity) of an optically-induced moir´e lattice is achieved, are determined by thePythagorean triples in the case of square sublattices or by another Diophantine equation,when the primitive cell of the sublattices is not square (see Methods). For all other rota-tion angles the structure is aperiodic albeit regular (i.e., it is not disordered). Importantly,changing the relative amplitudes of the sublattices allows to smoothly tune the shape ofthe lattice without affecting its rotational symmetry. In contrast to crystalline moir´e lat-tices [1–6], optical patterns are monolayer structures, i.e., both sublattices interfere in oneplane. As a consequence, light propagating in such media is described by a one-componentfield [see Eq. (1) below], rather than by the spinors characterizing electronic states in thetight-binding approximation applicable for material moir´e lattices [8, 9].In the paraxial approximation, the propagation of an extraordinary polarized light beamin a photorefractive medium with optically induced refractive index landscape is governedby the Schr¨odinger equation for the dimensionless signal field amplitude ψ ( r , z ) [30, 32]: i ∂ψ∂ z = − ∇ ψ + E I ( r ) ψ. (1)Here ∇ = ( ∂/∂x, ∂/∂y ); r = ( x, y ) is the radius-vector in the transverse plane, which is scaledto the wavelength λ = 632 . z is the propagationdistance scaled to the characteristic diffraction length 2 πn e λ ; n e is the refractive index of thehomogeneous crystal for extraordinary-polarized light, E > I ( r ) ≡ | p V ( r ) + p V ( S r ) | is the moir´e lattice induced by two ordinary polarizedmutually coherent periodic sublattices V ( r ) and V ( S r ) interfering in the ( x, y ) plane androtated by the angle θ with respect to each other (see Methods and [30] for details of theoptical induction technique); S = S ( θ ) is the operator of the two-dimensional rotation; p p are the amplitudes of the first and second sublattices, respectively. The numberof laser beams creating each sublattice V ( r ) depends on the desired lattice geometry. Theform in which the lattice intensity I ( r ) enters Eq. (1) is determined by the mechanism ofphotorefractive response.To visualize the formation of moir´e lattices it is convenient to associate a continuoussublattice V ( r ) with a discrete one with lattice vectors determined by the locations of theabsolute maxima of V ( r ). The resulting moir´e pattern inherits the rotational symmetry of V ( r ). At specific angles some nodes of different sublattices may coincide thereby leading totranslational symmetry of the moir´e patterns I ( r ) ≡ | p V ( r ) + p V ( S r ) | . Here we focuson square and hexagonal sublattices V ( r ), whose primitive translation vectors, in commen-surable phases, are illustrated by blue arrows, respectively, in Fig. 1 (the first and thirdcolumns) and in Fig. 4 (the first and second columns). The rotation angles at which theperiodicity of I ( r ) is achieved are determined by triples of positive integers ( a, b, c ) ∈ Z + related by a Diophantine equation characteristic for a given sublattice [31] (see Methods).First, we consider a Pythagorean lattice created by two square sublattices. For the rotationangles θ , such that cos θ = a/c and sin θ = b/c , where ( a, b, c ) is a Pythagorean triple, i.e., a + b = c , I ( r ) a is fully periodic (commensurable) moir´e lattice. Such angle is referredbelow as Pythagorean. For all other rotation non-Pythagorean angles θ , the lattice I ( r )is aperiodic (incommensurable). Figures 1(a)-(c) compare calculated I ( r ) patterns with p = p (first row) with lattices created experimentally in a biased SBN:61 photorefractivecrystal with dimensions 5 × ×
20 mm (third row) for different rotation angles indicatedon the panels. The lattice was created using optical induction technique, invented in [32]and first realized experimentally in [30]. The second row shows the respective discrete moir´elattices. Columns (a) and (c) show periodic lattices, while column (b) gives an example ofan aperiodic lattice. All results here and below were obtained for E = 7, which correspondsto a 8 × V/m dc electric field applied to the crystal. The amplitude of the first sublatticewas set to p = 1 in all cases, which corresponds to an average intensity I av ≈ . / cm .For such parameters, the actual refractive index modulation depth in the moir´e lattice is ofthe order of δn ∼ − . 5 IG. 1: (a)-(c) Moir´e lattices I ( r ) generated by two interfering square sublattices with p = p ,whose axes are mutually rotated by the angle indicated in each panel. First row: calculatedpatterns. Second row: schematic discrete representation of two rotated sublattices. Third row:experimental patterns observed at the output face of the crystal. The scale is the same for allexperimental images. Comparison of DOS calculated for moir´e lattice (top) and its periodic ap-proximation (bottom) at p = 0 . p = 0 . b = ... (see Supplementary Information), (e). (f) Band structures for periodic lat-tice approximating moir´e lattice at p = 0 . p = 0 . p = 1. An important feature of optical moir´e lattices that distinguishes them from their crystallinecounterparts, is that rotated sublattices have tunable amplitudes. The ratio p /p can bechanged without affecting the symmetry of the entire pattern, except for the case p = p ,when rotation symmetry changes, as discussed below. Thus, LDT in optical setting can bestudied either by varying the twist angle θ at fixed amplitudes p , , or by varying one orboth amplitudes of the sublattices, at a fixed angle θ .6rom the mathematical point of view, incommensurable lattices are almost periodic func-tions [35]. Like any irrational number can be approached by a rational one [36], any non-Pythagorean twist angle can be approached by a Pythagorean one with any prescribed accu-racy (as shown in Supplemental Information). Thus, any finite area of an incommensurablemoir´e lattice can be approached by a primitive effective cell of some periodic Pythagoreanlattice, and more accurate approximation requires larger primitive cell of the Pythagoreanlattice. This property (derived in Supplemental Information) is illustrated by quantitativesimilarities between densities of states (DOSs) calculated for an incommensurable latticeand its Pythagorean approximation and compared in Fig. 1(d,e) for different p /p ratios. Aremarkable property of Pythagorean lattices is extreme flattening of the higher bands thatoccurs when p /p ratio exceeds certain threshold, as illustrated in Fig. 1(f). The numberof flat bands rapidly grows with the size of the area of the primitive cell of the approxi-mating Pythagorean lattice. Thus, an incommensurable moir´e lattice can be viewed as thelarge-area limit of periodic Pythagorean lattices with extremely flat higher bands. Since flatbands support quasi-nondiffracting localized modes, the initially localized beam launchedinto such moir´e lattice will remain localized. This flat-band physics of moir´e lattices, earlierdiscussed for twisted bilayer graphene [8, 9, 34], allows us to predict beam localization abovesome threshold p /p value, whose physics is different from that of Anderson localizationin random media. Furthermore, flat bands support states, which are exponentially local-ized in the primitive cell. Such states can be well approximated by exponentially localizedtwo-dimensional Wannier functions [37] (see Fig. 2(c) and Supplementary Information).To elucidate the impact of the sublattice amplitudes and rotation angle θ on the lightbeam localization, we calculated the linear eigenmodes ψ ( r , z ) = w ( r ) e iβz , where β is thepropagation constant and w ( r ) is the mode profile, supported by the moir´e lattices. Tocharacterize the localization of the eigenmodes we use the integral form-factor χ defined by χ = U − (cid:82)(cid:82) | ψ | d r , where U = (cid:82)(cid:82) | ψ | d r is the energy flow (the integration is over thetransverse area of the crystal). Form-factor is inversely proportional to the mode width:the larger χ the stronger the localization. The dependence of the form-factor of the mostlocalized mode of the structure (mode with largest β ) on θ and p is shown in Fig. 2(a) (for7odes with lower β values these dependencies are qualitatively similar). One observes a sharpLDT above a certain threshold depth p LDT2 of the second sublattice, at fixed amplitude ofthe first sublattice: p = 1. This corroborates with strong band flattening of approximatingPythagorean lattice at p > p LDT2 depicted in Fig. 1(f). Below p LDT2 all modes are foundto be extended [Fig. 2(b)] and above the threshold, some modes are localized [Fig. 2(c)].Inset in Fig. 2(c) reveals exponential tails for p > p LDT2 from which localization length formost localized mode can be extracted. In Fig. 2(a) one also observes sharp delocalizationfor angles θ determined by the Pythagorean triples, when all modes are extended regardlessof p value.We observe in Fig. 2(a) that p LDT2 is practically independent of the rotation angle. Thisis explained by the fact that a large fraction of the power in a localized mode resides in thevicinity of a lattice maximum (i.e., at r = ). In an incommensurable phase I ( r ) < I ( )for all r (cid:54) = and the optical potential locally can be approximated by the Taylor expansionof E / [1 + I ( r )] with respect to r near the origin. Such expansion includes the rotationangle θ only in the fourth order (see Supplementary Information). Thus, locally, the opticalpotential can be viewed as almost isotropic.To study the guiding properties of the Pythagorean moir´e lattices experimentally and toexpose the two-dimensional LDT, we measured the diffraction outputs for beams propagatingin lattices corresponding to different rotation angles θ for fixed input position of the beam,centered or off-center. The diameter of the Gaussian beam focused on the input face ofthe crystal was about 23 µ m. Such a beam covers approximately one bright spot (channel)of the lattice profile. The intensity of the input beam was about 10 times lower than theintensity of the lattice-creating beam, I av , to guarantee that the beam does not distort theinduced refractive index and that it propagates in the linear regime. When a beam entersmoir´e lattice it either diffracts, if there are no localized modes, or it remains localized if suchmodes exist (notice that no averaging is used, because lattice is deterministic). ExtendedData Fig. 2 compares experimental and theoretical results for propagation dynamics in threedifferent regimes at fixed p = 1. In the incommensurable lattice at small p < p LDT2 oneobserves beam broadening and no traces of localization (top row). Localization takes place at8
IG. 2: (a) Form-factor (inverse width) of the eigenmodes with largest β versus rotation angle θ and versus amplitude of the second sublattice p at p = 1. The horizontal dashed line indicates thesublattice depth p LDT2 at which LDT occurs. The vertical dashed line shows one of the Pythagoreanangles θ p = arctan(3/4). Examples of mode profiles with largest β for p < p LDT2 (b) and p > p LDT2 (c). Insets show cuts of ln | ψ | distribution along the x and y axes. sufficiently high amplitude p > p LDT2 of the second sublattice in the incommensurable case,e.g., for p = 1 (middle row). However, at a twist angle θ corresponding to a commensurablemoir´e lattice, localization does not occur even for p = p = 1 due to the restoration of latticeperiodicity (bottom row). Simulations of propagation up to much larger distances beyondthe available sample length presented in Extended Data Fig. 3 fully confirm localization of9he beam in incommensurable lattice at any distance at p > p LDT2 and its expansion at p < p LDT2 . FIG. 3: Observed output intensity distributions illustrating LDT with increasing amplitude p of the second sublattice for rotation angle θ = arctan 3 − / = π/ θ = arctan(3 /
4) (central column). Insets show initialexcitation position, central for the left and central columns, and off-center for the right column.
Extended experimental evidence of LDT in the two-dimensional lattice is presented in Fig.3, where we compare output patterns for the low-power light beam in the incommensurable(tan θ = 3 − / , left and right columns for central and off-center excitations, respectively) andcommensurable (tan θ = 3 /
4, central column) moir´e lattices, tuning in parallel the amplitude p of the second sublattice. When p < p LDT2 (in Fig. 3 p LDT2 ≈ . I ( r ) in the vicinity of the excitation point. However, when p exceeds the10DT threshold, it is readily visible that diffraction is arrested for both central (left column)and off-center (right column) excitations and a localized spot is observed at the output overa large range of p values. In clear contrast, localization is absent for any p value in theperiodic lattice associated with the Pythagorean triple (central column). FIG. 4: First row: moir´e lattices produced by interference of two hexagonal patterns rotated bythe angle θ : p = 1 in the first, second, and fourth columns, while in the third column p = 0 . p = 1. The mutual rotation of two identical sublattices allows generation of commensurable andincommensurable moir´e patterns with sublattices of any allowed symmetry. To illustratethe universality of LDT, we induced hexagonal moir´e lattices (the technique of induction is11imilar to that used for single hexagonal photonic lattices [33]). For such lattices, the rotationangles producing commensurable patterns are given by the relation tan θ = b √ / (2 a + b ),where the integers a and b solve the Diophantine equation a + b + ab = c . Two examplesare presented in the first and second columns of Fig. 4. In such periodic structures, thesignal light beam experiences considerable diffraction for any amplitude of the sublattices,shown in the bottom row. To observe LDT one has to induce aperiodic structures. To suchend, we considered the rotation angle of 30 o . In such incommensurable case we did observeLDT by increasing the amplitude of the second sublattice, keeping the amplitude p fixed.Delocalized and localized output beams are shown in the lower panels of the third and fourthcolumns of Fig. 4. In the third column the ideal 6-fold rotation symmetry of the outputpattern is slightly distorted, presumably due to the intrinsic anisotropy of the photorefractiveresponse. At p = p the moir´e pattern acquires the 12-fold rotational symmetry (shown inthe fourth column of Fig. 4) as it was proposed in [12] as a model of a quasicrystal and issimilar to the twisted bilayer graphene quasicrystal reported in [6].In closing, we have created first photonic moir´e lattices by imprinting two mutually ro-tated square and hexagonal sublattices into the photorefractive crystal. These structureswith highly controllable parameters and symmetry have enabled systematic investigation ofevolution and localization of wavepackets. Namely, we report the first experimental demon-stration of two-dimensional LDT of a wavepacket in an optical setting. Our findings un-cover novel mechanism for wave localization in regular incommensurable systems based onthe physics of flat-band structures. LDT can occur according to different scenarios: viacommensurable-incommensurable transition and via tuning of relative depths of optically-induced lattices. In addition to new possibilities for control of light by light, i.e. controlof propagation paths, symmetry of diffraction patterns, diffraction rate of light beams, andformation of self-sustained excitations with new symmetries, photonic moir´e patterns allowsto study phenomena relevant to other areas of physics, particularly from condensed mattersystems, which are much harder to explore directly. Such lattices can be used to study therelations between conductivity/transport and symmetry of incommensurable patterns withlong-range order. Being tunable, optical moir´e patterns can be created in practically any12rbitrary configurations consistent with two-dimensional symmetry groups, thus allowingthe exploration of potentials that may not be easily produced in tunable form using materialmoir´e structures. 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P. W and F. Y acknowledge support from the NSFC (No. 61475101and No. 11690033). Y.V.K. and L.T. acknowledge support from the Severo Ochoa ExcellenceProgramme (SEV-2015-0522), Fundacio Privada Cellex, Fundacio Privada Mir-Puig, andCERCA/Generalitat de Catalunya.
Author contributions
P. W and Y. Z contributed equally to this work. All authorscontributed significantly to this work.
Competing interests
The authors declare no competing interests.16
ETHODS
Experimental setup.
The experimental setup is illustrated in Extended Data Fig. 1.A cw frequency-doubled Nd:YAG laser at wavelength λ = 532 nm is divided by a polariz-ing beam splitter into two polarization components, which are sent to Path a and Path b separately. Light in Path a is extraordinarily polarized and it is used to image the inducedpotential in the photorefractive crystal (see the third row of Fig. 1 in the main text). Lightin Path b is ordinarily polarized and it is used to write the desirable potential landscapein the photorefractive SBN:61 crystal with dimensions 5 × ×
20 mm and extraordinaryrefractive index n e = 2 . b is modulated by Masks 1 and 2 transforming this beam into superposition of tworotated periodic patterns. Their relative strength p /p , or, more precisely, the strength ofthe second lattice, as well as the twist angle θ are controlled by the polarizer-based Mask1 and amplitude Mask 2. The He-Ne laser with wavelength λ = 633 nm shown in path c provides extraordinarily polarized beam focused onto the front facet of the crystal, thatserves as a probe beam for studying light propagation in the induced potential. We recordthe output light intensity pattern by a CCD at the exit facet of the crystal after propagationdistance of 20 mm. Characteristics of moir´e lattices used in experiment.
In the experiments there havebeen used two types of moir´e lattices, whose characteristics are summarized in Extended DataTable I. In all the cases the center of rotation in the ( x, y ) plane was chosen coincident witha node of one of the sublattices.
DATA AVAILABILITY
The data that support the findings of this study are available from the correspondingauthor upon reasonable request. 17 xtended Data Fig. 1: Experimental setup. λ/
2, half-wave plate; PBS, polarizing beam splitter;SF, spatial filter; L, lens; BS, beam splitter; ID, iris diaphragm; M, mirror; P, Polarizer; SBN,strontium barium niobate crystal; CCD, charged-coupled device. Mask 2 is an amplitude mask toproduce two group of sub-lattices with a rotation angle θ , and Mask 1 is made of a polarizer film. EXTENDED DATA
Moir´e lattice I ( r ) Sublattice V ( r ) Diophantine equation tan θ Pythagorean cos(2 x ) + cos(2 y ) a + b = c b/a hexagonal (cid:80) n =1 cos [2( x cos θ n + y sin θ n )] a + b + ab = c √ b/ (2 a + b )Extended Data Tab. I: Characteristics of the moir´e lattices used in experiments. For hexagonallattices θ = 0, θ = 2 π/
3, and θ = 4 π/ xtended Data Fig. 2: Experimentally observed intensity distributions of the probe beam (color-surface plots) and corresponding theoretically calculated distributions (insets), at different prop-agation distances z , for tan θ = 3 − / , p = 0 .
1, which falls below the LDT point (top row),tan θ = 3 − / , p = 1, which falls above LDT point (middle row), and tan θ = 3 / p = 1 (bottomrow). The first two rows correspond to the incommensurable Pythagorean lattice shown in thecentral column of Fig. 1 of the main text. The third row corresponds to the commensurable latticeshown in the last column of Fig. 1 of the main text. xtended Data Fig. 3: (a),(b) Numerical simulations of the light beam propagation in the incom-mensurable moir´e lattice for central excitation, corresponding to top and middle rows of ExtendedData Fig. 2, but for larger distances, notably exceeding sample length. (c),(d) Similar numericalresults, but for off-center excitation position in moir´e lattice. Parameter p = 0 . p = 1 . θ = π/
6. In all cases Gaussian beam exciting a single siteof the potential is assumed.6. In all cases Gaussian beam exciting a single siteof the potential is assumed.